Hybrid functional studies of the oxygen vacancy in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mtext>TiO</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

The electronic and structural properties of the oxygen vacancy $({V}_{\text{O}})$ in rutile ${\text{TiO}}_{2}$ are studied using generalized Kohn-Sham theory with the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. The HSE approach corrects the band gap and allows for a proper description of defects with energy levels close to the conduction band. According to the HSE calculations, ${V}_{\text{O}}$ is a shallow donor for which the $+2$ charge state is lower in energy than the neutral and $+1$ charge states for all Fermi-level positions in the band gap. The formation energy of ${V}_{\text{O}}^{2+}$ is relatively low in $n$-type ${\text{TiO}}_{2}$ under O-poor conditions but it rapidly increases with the oxygen chemical potential. This is consistent with experimental observations where the electrical conductivity decreases with oxygen partial pressure.